User:Thierry Dugnolle/Wave-particle duality and the superposition principle

Wave-particle duality, the quantum superposition principle and the Born rule

One of the important consequences of the first postulate (the quantum superposition principle) when it is combined with the others, is the appearance of interference effects such as those which led us to wave-particle duality.^[1]

The quantum superposition principle is:

Physical states are represented by rays in Hilbert space. A Hilbert space is a kind of complex vector space; that is, if Φ and Ψ are vectors in the space (often called 'state vectors') then so is ${\displaystyle \xi }$Φ + ${\displaystyle \eta }$Ψ, for arbitrary complex numbers ${\displaystyle \xi ,\eta }$. ^[2] .

The vector Φ is also written ${\displaystyle |\phi \rangle }$ in Dirac notation. Two vectors ${\displaystyle |u\rangle }$ and ${\displaystyle |v\rangle }$ (different from the null vector) belong to the same ray if and only if there is a complex number ${\displaystyle \alpha }$ such that ${\displaystyle |u\rangle =\alpha |v\rangle }$.

The existence of a wave function for a particle is a direct consequence of the quantum superposition principle: since any particle can be in any localized state ${\displaystyle |x\rangle }$ (a localized state centered on ${\displaystyle x}$ with width ${\displaystyle dx}$) it can also be in a superposition of these states ${\displaystyle \sum _{x}\psi (x)|x\rangle }$, or ${\displaystyle \int \psi (x)|x\rangle dx=|\psi \rangle }$ (if ${\displaystyle dx}$ tends to zero) where ${\displaystyle \psi (x)}$ is the wave function (at a given time) of the particle:

The wave functions ${\displaystyle \psi (x)}$ that we have been using to describe physical states in wave mechanics should be considered as the set of components of an abstract vector ${\displaystyle \psi }$ known as the state vector.^[3]

${\displaystyle \psi (x)=\langle x|\psi \rangle }$ in Dirac notation. ${\displaystyle x}$ is a real number for a one-dimensional wave function, and a three-dimensional vector, for a three-dimensional wave function. In the latter case ${\displaystyle dx}$ is a three-dimensional box. A wave function at a given time is a state vector. A wave function at all times is the evolution of a state vector.

The Born rule states that the squared modulus ${\displaystyle |\psi (x)|^{2}}$of the normalized wave function ${\displaystyle \psi (x)}$ is the density probability of detection of the particle^[4]. (Normalized means that ${\displaystyle \int |\psi (x)|^{2}dx=1}$). This leads to wave-particle duality^[4][1]: the motion of a particle (the evolution of its state) is represented by a wave function^[5] (at all times) but the particle is always detected at a single place^[4].

A wave function can be a wave packet. Wave packets are waves which are localized. This means that their spreading in space, at a given time, is negligible (not measurable) at a distance. A wave packet can represent the evolution of the state of a quantum particle^[6]:

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  This wave packet represents the motion of a quantum particle ^[6]. The motion of a particle can also be represented by two wave packets (see next animation) through quantum superposition, or by any other wave function.
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  These two wave packets represent the motion of a single particle. This shows that a quantum particle can interfere with itself.
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  The Born rule in action: simulated experiment of the detection of particles in an interference pattern.

1. Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, chapter III, section E
2. Steven Weinberg, The Quantum theory of fields, vol.I, p.49
3. Steven Weinberg, Lectures on quantum mechanics, p.53
4. Born, Max (1926). "Zur Quantenmechanik der Stoßvorgänge". Zeitschrift für Physik. 37 (12): 863–867. Bibcode:1926ZPhy...37..863B. doi:10.1007/BF01397477. S2CID 119896026. Reprinted in English translation as Born, Max (1983). "On the quantum mechanics of collisions". In Wheeler, J. A.; Zurek, W. H. (eds.). Quantum Theory and Measurement. Princeton University Press. pp. 52–61. ISBN 978-0-691-08316-2.
5. Cohen-Tannoudji, Diu and Laloë, Quantum mechanics, chapter I, section C
6. Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, complement G-I